Past Number Theory Seminar

12 May 2016
16:00
Norbert A’Campo
Abstract
An implicite definition for the hyperbolic plane $H=H_I$ is in: ${\rm Spec}(\mathbb{R}[X]) = H_I \cup \mathbb{R}$. All geometric hyperbolic features will follow from this definition in an elementary way.
 
A second definition is $H=H_J=\{J \in {\rm End}(R^2) \mid J^2=-Id, dx \wedge dy(u,Ju) \geq 0 \}$. Working with $H=H_J$ allows to prove rather directly main theorems about Riemann surfaces.
  • Number Theory Seminar
5 May 2016
16:00
Giovanni Rosso
Abstract

In a recent work Andreata, Iovita, and Pilloni constructed the eigenvariety for cuspidal Siegel modular forms. This eigenvariety has the expected dimension (the genus of the Siegel forms) but it parametrizes only cuspidal forms. We explain how to generalize the construction to the non-cuspidal case. To be precise, we introduce the notion of "degree of cuspidality" and we construct an eigenvariety that parametrizes forms of a given degree of cuspidability. The dimension of these eigenvarieties depends on the degree of cuspidality we want to consider: the more non-cuspidal the forms, the smaller the dimension. This is a joint work with Riccardo Brasca.

  • Number Theory Seminar
28 April 2016
16:00
Andrew Ranicki
Abstract

The talk will be based on some of the material in the joint survey with Etienne Ghys

"Signatures in algebra, topology and dynamics"

http://arxiv.org/abs/1512.092582

In the 19th century Sturm's theorem on the number of roots of a real polynomial motivated Sylvester to define the signature of a quadratic form. In the 20th century the classification of quadratic forms over algebraic number fields motivated Witt to introduce the "Witt groups" of stable isomorphism classes of quadratic forms over arbitrary fields. Still in the 20th century the study of high-dimensional topological manifolds with nontrivial fundamental group motivated Wall to introduce the "Wall groups" of stable isomorphism classes of quadratic forms over arbitrary rings with involution. In our survey we interpreted Sturm's theorem in terms of the Witt-Wall groups of function fields. The talk will emphasize the common thread running through this developments, namely the notion of the localization of a ring inverting elements. More recently, the Cohn localization of inverting matrices over a noncommutative ring has been applied to topology in the 21st century, in the context of the speaker's algebraic theory of surgery.

 

  • Number Theory Seminar
10 March 2016
16:00
Igor Wigman
Abstract

We study the number of nodal domains of toral Laplace eigenfunctions. Following Nazarov-Sodin's results for random fields and Bourgain's de-randomisation procedure we establish a precise asymptotic result for "generic" eigenfunctions. Our main results in particular imply an optimal lower bound for the number of nodal domains of generic toral eigenfunctions.

  • Number Theory Seminar
3 March 2016
16:00
Neil Dummigan
Abstract

I will describe how the moduli of various congruences between Hecke eigenvalues of automorphic forms ought to show up in ratios of critical values of $\text{GSP}_2 \times \text{GL}_2$ L-functions. To test this experimentally requires the full force of Farmer and Ryan's technique for approximating L-values given few coefficients in the Dirichlet series.

  • Number Theory Seminar
25 February 2016
16:00
Victor Beresnevich
Abstract

I will discuss the notion of badly approximable points and recent progress and problems in this area, including Schmidt's conjecture, badly approximable points on manifolds and real numbers badly approximable by algebraic numbers.

  • Number Theory Seminar
4 February 2016
16:00
Damian Rössler
Abstract

We shall describe a new proof of the Mordell-Lang conjecture in positive characteristic, in the situation where the variety under scrutiny is a smooth subvariety of an abelian variety. 
Our proof is based on the theory of semistable sheaves in positive characteristic, in particular on  Langer's theorem that the Harder-Narasimhan filtration of sheaves becomes strongly semistable after a finite number of iterations of Frobenius pull-backs. Our proof produces a numerical upper-bound for the degree of the finite morphism from an isotrivial variety appearing in the statement of the Mordell-Lang conjecture. This upper-bound is given in terms of the Frobenius-stabilised slopes of the cotangent bundle of the variety.

  • Number Theory Seminar
28 January 2016
16:00
David Loeffler
Abstract

Iwasawa theory is a powerful technique for relating the behaviour of arithmetic objects to the special values of L-functions. Iwasawa originally developed this theory in order to study the class groups of number fields, but it has since been generalised to many other settings. In this talk, I will discuss some new results in the Iwasawa theory of the symmetric square of a modular form. This is a joint project with Sarah Zerbes, and the main tool in this work is the Euler system of Beilinson-Flach elements, constructed in our earlier works with Kings and Lei.

  • Number Theory Seminar

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